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Probability Rules

P robability is the likelihood of the occurrence of an event where the random outcome of an event cannot be predicted with certainty.
Statistical experiment or observation is an activity in which the results or outcome are definite, but they are not random.
An event of the sample space is defined as the basis structure of the outcomes, which cannot be further classified, where a sample space is the total of all events in a statistical experiment. 

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Probability Rules:
1.    The probability of any event always lies between zero and one. That is, for any event A, 0 $\leq $ P(A) $\leq $ 1.
2.    The total of all the event’s probabilities must be equal to one.
3.    The complement of an event A, has all the outcomes that are not a part of event A, and hence, P(A’) = 1 – P(A) or P (not A) or simply read as A compliment.
4.    The two events are known as mutually exclusive or disjoint events if they have no outcome common to each other. That is, their intersection will come out to be an empty set.
5.    The union of two events, A and B is made up of all the outcomes jointly together, in the complete sample space, that are in A or B, or both.
6.    The intersection of two events, A and B is made up of all the outcomes that are common to each other, in the complete sample space, that are in both A and B.
7.    Addition Rule: The probability of union of two events A and B is given by: P (A union B) = P(A) + P(B) – P(A ∩ B). If A and B are mutually exclusive then, P(A union B) = P(A) + P(B).
8.    Two events are said to be independent of each other if outcome of one event does not change the probability of other event.
9.    Conditional probability is defined as the probability of an event A, given that the event B has already occurred, and is calculated as P(A|B) = $\frac{P(A∩B)}{P(B)}$.
10.    Multiplication Rule: The probability of intersection of two events is given by: P(A ∩ B) = P(A|B) * P(B)

Solved Examples:
Example 1: Two dice are tossed. Find the probability of getting an even number on the first die or a total of 8
Solution: Here S = {1, 2, 3, 4, 5, 6} x {1, 2, 3, 4, 5, 6}, that is total number of outcome would be 6 * 6 = 36.
Let A = getting an even number on first die and
B = sum of points obtained on the two equals 8, then
A = {2, 4, 6} x {1, 2, 3, 4, 5, 6} has 3 * 6 = 18 outcomes and
B = {(2, 6), (6, 2), (3, 5), (5, 3), (4, 4)} has 5 outcomes
Hence, P(A) = $\frac{18}{36}$, P(B) = $\frac{5}{36}$

Example 2: Given P(A) = 0.30, P(B) = 0.60, P(A and B) = 0.25, find P (A or B)
Solution: By applying the rule P (A or B) = P (A) + P (B) – P(A and B) we will get,
P (A or B) = 0.30 + 0.60 – 0.25 = 0.65 is the answer.

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